Extreme Conditional Tail Moment Estimation under Serial Dependence

A wide range of risk measures can be written as functions of conditional tail moments (CTMs) and value-at-risk (VaR), for instance the expected shortfall (ES). In this paper, we derive joint central limit theory for semi-parametric estimates of CTMs, including in particular ES, at arbitrarily small risk levels. We also derive confidence corridors for VaR at different levels far out in the tail, which allows for simultaneous inference. We work under a semi-parametric Pareto-type assumption on the distributional tail of the observations and only require an extremal-near epoch dependence assumption on the serial dependence. In simulations, our semi-parametric ES estimate is often shown to be more accurate in terms of mean absolute deviation than extant non- and semi-parametric estimates. An empirical application to the extreme swings in Volkswagen log-returns during the failed takeover attempt by Porsche illustrates the proposed methods.